cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Graeme McRae

Graeme McRae's wiki page.

Graeme McRae has authored 120 sequences. Here are the ten most recent ones:

A299700 Squarefree part of 1!*2!*3!*...*n!: The product of factorials one through n divided by its largest square divisor.

Original entry on oeis.org

1, 2, 3, 2, 15, 3, 105, 6, 105, 15, 1155, 5, 15015, 70, 1001, 70, 17017, 35, 323323, 7, 138567, 154, 3187041, 231, 3187041, 6006, 1062347, 858, 30808063, 715, 955049953, 1430, 260468169, 12155, 9116385915, 12155, 337306278855, 461890, 8648878945, 46189, 354604036745, 1939938, 15247973580035, 176358
Offset: 1

Views

Author

Graeme McRae, Feb 17 2018

Keywords

Comments

Smallest number such that a(n)*1!*2!*3!*...*n! is a square.
If n is even, a(2n) = A055204(n).
If n is odd and evil (A129771) then a(2n) = A055204(n)/2.
If n is odd and odious (A092246) then a(2n) = 2*A055204(n).

Examples

			1!*2!*3!*4!*5! = 2^8 * 3^3 * 5^1 so a(5) = 3*5 = 15.

Links

Crossrefs

Cf. A000178, A007913, A055204, A092246, A129771.

Programs

  • Mathematica
    Nest[Append[#, {#, Sqrt[#] /. (c_: 1) a_^(b_: 0) :> (c a^b)^2} &[#[[-1, 1]]*Length[# + 1]!]] &, {{1, 1}}, 44][[All, -1]] (* Michael De Vlieger, Feb 17 2018, after Bill Gosper at A007913 *)
f[n_] := Block[{m = BarnesG[n +2], p = 2}, While[p < n, While[ Mod[m, p^2] == 0, m/=p^2]; p = NextPrime@ p]; m]; Array[f, 42] (* Robert G. Wilson v, Feb 18 2018 *)
  • PARI
    a(n) = core(prod(k=1, n, k!)); \\ Michel Marcus, Feb 17 2018

Formula

a(n) = A007913(A000178(n)). - Michel Marcus, Feb 17 2018

A200730 Smallest nontrivial positive power x such that the number of odd powers (i.e., odd base) not exceeding x exceeds by n the number of even powers not exceeding x.

Original entry on oeis.org

1, 5041, 1601613, 1221611509, 355183455415293
Offset: 1

Views

Author

Graeme McRae, Nov 21 2011

Keywords

Examples

			a(3)=1601613 because there are 699 odd powers not larger than 1601613, and 696 even powers not larger than 1601613, and 699 - 696 = 3.

Crossrefs

Cf. A200729.

A200729 Smallest nontrivial positive power x such that the number of even powers (i.e., even base) not exceeding x exceeds by n the number of odd powers not exceeding x.

Original entry on oeis.org

4, 8, 216, 2116, 551368, 47444544, 17649109134656
Offset: 0

Views

Author

Graeme McRae, Nov 21 2011

Keywords

Comments

A nontrivial power is a^b where a,b are integers and b>1.

Examples

			a(2)=216 because there are 11 even powers not larger than 216 and 9 odd powers not larger than 216, and 9 - 7 = 2.

Crossrefs

Cf. A200730.

A137939 Number of 5-way intersections in the interior of a regular 6n-gon.

Original entry on oeis.org

0, 0, 54, 24, 180, 216, 546, 336, 648, 720, 990, 936, 1404, 2352, 1890, 1824, 2448, 2592, 3078, 3720, 4284, 3960, 4554, 4464, 5400, 5616, 6318, 7896, 7308, 7560, 8370, 8256, 9504, 9792, 11550, 10584, 11988, 12312, 13338, 14640, 14760, 17640, 16254, 16104, 17820, 18216, 19458, 19296, 22344, 21600
Offset: 1

Views

Author

Graeme McRae, Feb 23 2008

Keywords

Comments

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

Examples

			a(3) = 54 because there are 54 points in the interior of an 18-gon at which exactly five diagonals meet.

Links

Crossrefs

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon..
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.
Cf. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon.

Formula

a(n) = A101365(6*n). - Seiichi Manyama, Jul 20 2024

Extensions

More terms from Seiichi Manyama, Jul 20 2024

A137938 Number of 4-way intersections in the interior of a regular 6n-gon.

Original entry on oeis.org

0, 12, 54, 264, 420, 396, 1134, 1200, 1296, 3780, 2310, 2520, 3276, 3612, 4050, 5088, 5712, 5724, 7182, 11400, 9072, 9372, 10626, 11088, 12600, 13260, 14094, 15960, 17052, 23220, 19530, 20928, 21384, 23052, 26250, 25704, 27972, 28956, 30186, 39600, 34440, 34524
Offset: 1

Views

Author

Graeme McRae, Feb 23 2008

Keywords

Comments

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

Examples

			a(3)=54 because there are 54 points in the interior of an 18-gon at which exactly four diagonals intersect.

Links

Crossrefs

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.
Cf. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon

Formula

a(n) = A101364(6*n). - Seiichi Manyama, Jul 20 2024

Extensions

More terms from Seiichi Manyama, Jul 20 2024

A130733 Numbers whose square can be expressed as a+b*c, with a,b,c in geometric sequence.

Original entry on oeis.org

3, 102, 130, 312, 759, 2496, 2706, 3465, 6072, 6111, 8424, 14004, 16005, 36897, 37156, 92385, 98640, 112032, 117708, 128040, 351260, 378108, 740050, 1346400, 1371900, 1898130, 3998607, 5986575, 6082065, 6631596, 6741214, 7692804
Offset: 1

Views

Author

Graeme McRae, Jul 05 2007

Keywords

Comments

This sequence was inspired by a puzzle question that asked for all squares under a trillion that can be expressed as either a+bc or b+ac where a,b,c are in increasing geometric progression.

Examples

			a(1)=3 because 3^2=1+2*4 and 1,2,4 are in geometric sequence
a(2)=102 because 102^2=36+72*144 and 36,72,144 are in geometric sequence
a(3)=130 because 130^2=25+75*225 and 25,75,225 are in geometric sequence
a(4)=312 because 312^2=8+92*1058 and ...
a(5)=759 because 759^2=81+360*1600
a(6)=2496 because 2496^2=512+1472*4232
a(7)=2706 because 2706^2=1936+2420*3025
a(8)=3465 because 3465^2=1225+2450*4900
a(9)=6072 because 6072^2=5184+5760*6400
a(10)=6111 because 6111^2=3969+5292*7056
a(11)=8424 because 8424^2=5832+7452*9522
a(12)=14004 because 14004^2=432+4392*44652
a(13)=16005 because 16005^2=1089+6534*39204
a(14)=36897 because 36897^2=21609+30870*44100
a(15)=37156 because 37156^2=12544+25872*53361
a(16)=92385 because 92385^2=50625+75600*112896
a(17)=98640 because 98640^2=50625+78975*123201
a(18)=112032 because 112032^2=27648+70272*178608
a(19)=117708 because 117708^2=41616+83232*166464
a(20)=128040 because 128040^2=69696+104544*156816
a(21)=351260 because 351260^2=67600+202800*608400
a(22)=378108 because 378108^2=314928+355752*401868
a(23)=740050 because 740050^2=521284+658464*831744
No other numbers smaller than a million have squares that can be expressed this way.
Contribution from _Donovan Johnson_, Jul 30 2010: (Start)
a(24)=1346400 because 1346400^2=135000+625500*2898150
a(25)=1371900 because 1371900^2=10000+266000*7075600
a(26)=1898130 because 1898130^2=6084+279864*12873744
a(27)=3998607 because 3998607^2=1413721+2827442*5654884
a(28)=5986575 because 5986575^2=1157625+3461850*10352580
a(29)=6082065 because 6082065^2=4348377+5438466*6801828
a(30)=6631596 because 6631596^2=1944+440532*99829446
a(31)=6741214 because 6741214^2=334084+2476152*18352656
a(32)=7692804 because 7692804^2=444528+2974104*19898172
(End)

Extensions

Added word: 'increasing'. The original puzzle was expressed as a modulo operation, the expression was 'remainder + quotient * divisor', where the remainder is necessarily smaller than the divisor, implying an increasing sequence. Counterexample if 'increasing' is not specified: a=8, b=4, c=2. a+b*c = 16 = 4^2; 4 is not in sequence A130733 - James Cunnane (james.cunnane(AT)gmail.com), Jun 29 2010
a(24)-a(32) from Donovan Johnson, Jul 30 2010

A132131 "Punctual Bird" numbers n with the additional property that n-1 is not a Punctual Bird (cf. A131881).

Original entry on oeis.org

1, 13, 22, 24, 33, 35, 44, 46, 55, 57, 66, 68, 77, 79, 88, 90, 100, 102, 113, 124, 133, 143, 153, 163, 173, 183, 193, 203, 224, 235, 244, 254, 264, 274, 284, 294, 304, 335, 346, 355, 365, 375, 385, 395, 405, 446, 457, 466, 476, 486, 496, 506, 557, 568, 577, 587
Offset: 1

Views

Author

Graeme McRae, Aug 11 2007

Keywords

Comments

Punctual Birds (A131881) are all numbers k with A132131(n) <= k < A132132(n) for some n Early Birds (A116700) are all numbers k with A132132(n) <= k < A132131(n+1) for some n

Examples

			a(1)=1 because 1 is the first Punctual Bird.
a(2)=13 because 1-11 are Punctual Birds and 12 is not a Punctual Bird.
a(3)=22 because 13-20 are Punctual Birds and 21 is not a Punctual Bird.

Links

Crossrefs

Cf. A033307, A117804, A116700, A131881, A132132, A132133.

A132132 "Early Bird" numbers n such that n-1 is not an Early Bird (cf. A116700).

Original entry on oeis.org

12, 21, 23, 31, 34, 41, 45, 51, 56, 61, 67, 71, 78, 81, 89, 91, 101, 110, 121, 131, 141, 151, 161, 171, 181, 191, 201, 210, 231, 241, 251, 261, 271, 281, 291, 301, 310, 341, 351, 361, 371, 381, 391, 401, 410, 451, 461, 471, 481, 491, 501, 510, 561, 571, 581
Offset: 1

Views

Author

Graeme McRae, Aug 11 2007

Keywords

Comments

Punctual Birds (A131881) are all numbers k with A132131(n) <= k < A132132(n) for some n Early Birds (A116700) are all numbers k with A132132(n) <= k < A132131(n+1) for some n

Examples

			a(1)=12 because 12 is the first Early Bird.
a(2)=21 because 21 is the second Early Bird.
a(3)=23 because 23 is the third Early Bird.
a(4)=31 because 31 is the fourth Early Bird.
a(5)=34 because 31-32 are Early Birds and 33 is not an Early Bird.

Links

Crossrefs

Cf. A033307, A117804, A116700, A131881, A132131, A132133.

A132135 Number of base-3 "Punctual Birds" with n base-3 digits.

Original entry on oeis.org

2, 3, 4, 10, 20, 51, 133, 359, 975, 2700, 7506, 20962, 58968, 166170
Offset: 1

Views

Author

Graeme McRae, Aug 11 2007

Keywords

Examples

			a(2)=3 because 3 of the six 2-digit base-3 numbers appear in their natural order in the string of concatenated base-3 numbers. These 3 numbers are 3, 4 and 6 (10, 11 and 20 base-3).

Crossrefs

Cf. A132134.

A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 15, 18, 27, 29, 30, 32, 33, 35, 42, 45, 54, 60, 81, 83, 86, 87, 89, 92, 95, 96, 98, 99, 101, 104, 105, 107, 123, 126, 135, 141, 153, 162, 243, 245, 248, 249, 251, 252, 254, 257, 258, 260, 261, 263, 266, 267, 269, 275, 276, 278, 285, 287, 288
Offset: 1

Views

Author

Graeme McRae, Aug 11 2007

Keywords

Examples

			a(5)=6 because 6 (20, base 3) is the fifth number that appears first in its "natural" place in the string of concatenated base-3 numbers.

Crossrefs

Cf. A033307, A117804, A116700, A131881, A132131, A132132, A132133, A132135.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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